For all \(\alpha _1,\alpha _2>0\) , we can choose \((u,v)\in S(a_1)\times S(a_2)\)  such that (see [1]})
\(\Vert \nabla u\Vert ^2_2>\frac{2\alpha _1}{3}\Vert u\Vert ^3_3\quad \text{and}\quad \Vert \nabla v\Vert ^2_2 >\frac{2\alpha _2}{3}\Vert v\Vert ^3_3.\) 
